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Intersection Theorem for DG-modules over non-positive DG-rings

Part of: Homological methods Theory of modules and ideals Commutative algebra: Homological methods

Published online by Cambridge University Press:  12 December 2025

Xiaoyan Yang
Xiaoyan Yang*
Affiliation:
School of Science, Zhejiang University of Science and Technology, Hangzhou, China (yangxy@zust.edu.cn)
*
*Corresponding author.
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Abstract

By extending some basic results about cohomological dimension of tensor products to non-positive DG-rings, the Intersection Theorem for DG-modules is examined over commutative noetherian local DG-rings with bounded cohomology. Some applications are provided. The first is to improve the DG-setting of the amplitude inequality in [Forum Math. 22 (2010) 941–948]. The second is to show Minamoto’s conjecture in [Israel J. Math. 242 (2021) 1–36]. The third is to obtain the DG-version of the Vasconcelos conjecture about Gorenstein rings.

Keywords

cohomological dimensiondepthIntersection Theoremamplitude inequalityCohen-Macaulay DG-ringGorenstein DG-ring

MSC classification

Primary: 13D22: Homological conjectures (intersection theorems)
Secondary: 13C15: Dimension theory, depth, related rings (catenary, etc.) 16E45: Differential graded algebras and applications

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 18
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

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References

Avramov, L. L., Iyengar, S. B. and Neeman, A.. Big Cohen-Macaulay modules, morphisms of perfect complexes and intersection theorems in local algebra. Doc. Math. 23 (2018), 1601–1619.10.4171/dm/654CrossRefGoogle Scholar
Bass, H.. On the ubiquity of Gorenstein rings. Math. Z. 82 (1963), 8–28.10.1007/BF01112819CrossRefGoogle Scholar
Bird, I., Shaul, L., Sridhar, P. and Williamson, J.. Finitistic dimension over commutative DG-rings. Math. Z. 309 (2025), 1–29.10.1007/s00209-024-03617-2CrossRefGoogle Scholar
Christensen, L. W., Foxby, H. -B. and Holm, H.. Derived Category Methods in Commutative Algebra. Springer Monographs in Mathematics. Switzerland: Springer Nature (2024), https://doi.org/10.1007/978-3-031-77453-9.Google Scholar
Dibaei, M. T. and Yassemi, S.. Generalized local cohomology and the Intersection Theorem. Comm. Algebra. 33 (2005), 899–908.10.1081/AGB-200051162CrossRefGoogle Scholar
Foxby, H. -B. and Iyengar, S. B.. Depth and amplitude for unbounded complexes. Contemp. Math. 331 (2003), 119–137.10.1090/conm/331/05906CrossRefGoogle Scholar
Frankild, A. and Jøgensen, P.. Homological identities for differential graded algebras. J. Algebra. 265 (2003), 114–135.10.1016/S0021-8693(03)00025-5CrossRefGoogle Scholar
Hochster, M.. Topics in the homological theory of modules over commutative rings. Amer. Math. Soc. 24 (1975).Google Scholar
Iversen, B.. Amplitude inequalities for complexes. Ann. Sci. ÉCole Norm. Sup. 10 (1977), 547–558.10.24033/asens.1336CrossRefGoogle Scholar
Iyengar, S. B.. Depth for complexes and intersection theorems. Math. Z. 230 (1999), 545–567.10.1007/PL00004705CrossRefGoogle Scholar
Iyengar, S. B., Ma, L. Q., Schwede, K. and Walker, M. E.. Maximal Cohen-Macaulay complexes and their uses: a partial survey. In: Commutative Algebra, pp. 475–500 (Switzerland: Springer Nature, 2021).10.1007/978-3-030-89694-2_15CrossRefGoogle Scholar
Jorgensen, P.. Amplitude inequalities for differential graded modules. Forum Math. 22 (2010), 941–948.10.1515/forum.2010.049CrossRefGoogle Scholar
Minamoto, H.. Homological identities and dualizing complexes of commutative differential graded algebras. Israel J. Math. 242 (2021), 1–36.10.1007/s11856-021-2095-3CrossRefGoogle Scholar
Minamoto, H.. Resolutions and homological dimensions of DG-modules commutative DG-algebras. Israel J. Math. 245 (2021), 409–454.10.1007/s11856-021-2230-1CrossRefGoogle Scholar
Peskine, C. and Szpiro, L.. Dimension projective finie et cohomologie locale. Inst. Hautes ÉTudes Sci. Publ. Math. 42 (1973), 47–119.10.1007/BF02685877CrossRefGoogle Scholar
Piepmeyer, G. and Walker, M. E.. A new proof of the New Intersection Theorem. J. Algebra. 322 (2009), 3366–3372.10.1016/j.jalgebra.2007.09.015CrossRefGoogle Scholar
Roberts, P.. Two applications of dualizing complexes over local rings. Ann. Sci. ÉCole Norm. Sup. 9 (1976), 103–106.10.24033/asens.1305CrossRefGoogle Scholar
Sharif, T. and Yassemi, S.. Special homological dimensions and Intersection Theorem. Math. Scand. 96 (2005), 161–168.10.7146/math.scand.a-14950CrossRefGoogle Scholar
Shaul, L.. The twisted inverse image pseudofunctor over commutative DG rings and perfect base change. Adv. Math. 320 (2017), 279–328.10.1016/j.aim.2017.08.041CrossRefGoogle Scholar
Shaul, L.. Injective DG-modules over non-positive DG-rings. J. Algebra. 515 (2018), 102–156.10.1016/j.jalgebra.2018.07.040CrossRefGoogle Scholar
Shaul, L.. Completion and torsion over commutative DG rings. Israel J. Math. 232 (2019), 531–588.10.1007/s11856-019-1866-6CrossRefGoogle Scholar
Shaul, L.. The Cohen-Macaulay property in derived commutative algebra. Trans. Amer. Math. Soc. 373 (2020), 6095–6138.10.1090/tran/8099CrossRefGoogle Scholar
Shaul, L.. Koszul complexes over Cohen-Macaulay rings. Adv. Math. 386 (2021), 107806.10.1016/j.aim.2021.107806CrossRefGoogle Scholar
Shaul, L.. Sequence-regular commutative DG-rings. J. Algebra. 647 (2024), 400–435.10.1016/j.jalgebra.2024.02.034CrossRefGoogle Scholar
Vasconcelos, W. V.. Divisor Theory in Module Categories. (North-Holl and Publishing Co, Amsterdam, 1974).Google Scholar
Yang, X. Y. and Li, Y. J.. Local Cohen-Macaulay DG-modules. Appl. Categor. Struct. 31 (2023), 1–13.10.1007/s10485-022-09703-yCrossRefGoogle Scholar
Yang, X. Y. and Wang, L.. Homological invariants over non-positive DG-rings. J. Algebra Appl. (2020), 2050153.10.1142/S0219498820501534CrossRefGoogle Scholar
Yassemi, S.. Generalized section functors. J. Pure Appl. Algebra. 95 (1994), 103–119.10.1016/0022-4049(94)90121-XCrossRefGoogle Scholar
Yekutieli, A.. Duality and tilting for commutative DG-rings. (2016) arXiv:1312.6411v4.Google Scholar